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Duffin--Schaeffer examples, real residue systems, and Bohr-set primes

Stefan M. Hesseling, Felipe A. Ramirez

Abstract

We prove the following generalization of a well-known result of Duffin and Schaeffer: For any given countable sets $Y \subset\mathbb{R}$ and $Z\subset\mathbb{R}\setminus\operatorname{span}_\mathbb{Q}(\{1\}\cup Y)$, there exist functions $ψ$ such that the set of inhomogeneously $ψ$-approximable numbers has zero measure or full measure, according as the inhomogeneous parameter lies in $Y$ or $Z$. The proof uses an analogue of residue systems where the residues can take arbitrary real values, and it also requires information about the distribution of primes lying in Bohr sets. We extend a theorem of Rogers to the more general real residues setting, and we extend Dirichlet's theorem for prime numbers lying in arithmetic progressions to prime numbers lying in Bohr sets. We also prove that circle rotations equidistribute when sampled along such primes, provided the rotation angle is rationally independent of the Bohr set parameter, generalizing a theorem of Vinogradov.

Duffin--Schaeffer examples, real residue systems, and Bohr-set primes

Abstract

We prove the following generalization of a well-known result of Duffin and Schaeffer: For any given countable sets and , there exist functions such that the set of inhomogeneously -approximable numbers has zero measure or full measure, according as the inhomogeneous parameter lies in or . The proof uses an analogue of residue systems where the residues can take arbitrary real values, and it also requires information about the distribution of primes lying in Bohr sets. We extend a theorem of Rogers to the more general real residues setting, and we extend Dirichlet's theorem for prime numbers lying in arithmetic progressions to prime numbers lying in Bohr sets. We also prove that circle rotations equidistribute when sampled along such primes, provided the rotation angle is rationally independent of the Bohr set parameter, generalizing a theorem of Vinogradov.
Paper Structure (13 sections, 11 theorems, 99 equations)

This paper contains 13 sections, 11 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Inhomogeneous Duffin--Schaeffer-type examples
  3. Real residue systems
  4. Distribution of primes lying in Bohr sets
  5. Further discussion and questions
  6. Proof of Theorem \ref{['prescription']}
  7. Real residue systems
  8. Distribution of primes lying in Bohr sets
  9. Equidistribution of toral translations
  10. Equidistribution along prime times
  11. Dirichlet's prime number theorem for Bohr sets
  12. Equidistribution along primes in Bohr sets
  13. Proof of þ \ref{['modestmeasureofunions']}

Key Result

Theorem 1.1

þ Let $Y,Z \subseteq\mathbb{R}$ be countable sets such that $Z\subseteq\mathbb{R}\setminus\mathop{\mathrm{span}}\nolimits_\mathbb{Q}(\lbrace1\rbrace\cup Y)$. Then there exists $\psi:\mathbb{N}\to \mathbb{R}_{\geqslant 0}$ such that for every $y\in Y$ and $z\in Z$.

Theorems & Definitions (23)

  • Theorem 1.1: Duffin--Schaeffer examples with prescribed behavior
  • Theorem 1.2: Rogers's theorem for real residues
  • Remark
  • Theorem 1.3: Dirichlet's prime number theorem for Bohr sets
  • Remark
  • Theorem 1.4: Equidistribution along Bohr-set primes
  • Remark
  • Proposition 2.1
  • proof : Proof of þ \ref{['prescription']} modulo þ \ref{['modestmeasureofunions']}
  • proof : Proof of þ \ref{['thm:rogersext']}
  • ...and 13 more